352 Journal of International Council on Electrical Engineering Vol. 1, No. 3, pp. 352~358, 2011
On-line Estimation of Power System Low Frequency Oscillatory Modes in Large Power Systems Chairerg Jakpattanajit *, Naebboon Hoonchareon † and Akihiko Yokoyama ** Abstract – In order to evaluate the small signal stability of power system, eigenvalue analysis is an important method to implement. However, there are some disadvantages for example it cannot analyze in on-line condition and is effective for calculating in limited operation conditions. This paper aims to investigate the performance of online oscillation mode estimation techniques for large power systems. Autoregressive and moving average (ARMA model) with least-square and kalman filter algorithms are employed and compared. The performance of both algorithms is tested in term of calculation time and mean square error. The results present ARMA model with both algorithms can provide the satisfied performance. However, ARMA model with kalman filter algorithm has more advantages such as faster calculation time and more accurate calculation results. Thus, this algorithm can be utilized in online oscillation mode estimation and can be used in design the control system for enhancing the damping in power system. Keywords: Power oscillation modes, Modal analysis, ARMA model, Least-square, Kalman filter
1. Introduction In modern large-scale power systems which contain remote generation and long distance power transmission, the low-frequency oscillation has been a serious problem. The significant method which is always employed in evaluating the dynamic performance and assisting in design controllers of power system is eigenvalue analysis [1][2]. This method consists of the following two steps. The first step is deriving a linear time-invariant model from a nonlinear power system by using small signal technique around certain operated point. The second step is finding eigenvalues of the nominal linear system and evaluating the small signal stability of the linearized power system. However, this method is impractical for a large-scale power system due to the complexity in solving large matrix in linear equation. Thus, the system identification method is implemented to reduce the time-consuming and complicated calculations. Two widely used methods are prony method [3] and ARMA model. Both of them were developed to estimate a subset of the eigenvalues of the large-scale power system from time signals measurement and do not require the system parameters [4]. As for prony †
Corresponding Author: Department of Electrical Engineering, Chulalongkorn University, Thailand (
[email protected]) * Department of Electrical Engineering, Chulalongkorn University, Thailand ** Department of Advanced Energy, The University of Tokyo, Japan Received: January 27, 2011; Accepted: June 14, 2011
method, the signals which are used in the calculations must be stimulated by large disturbances such as generation tipping and severe short circuits. However, the prony method always provide poor, yield biased, and sensitive calculation results of oscillation modes when the measured signals have low signal to noise ratio. The performances of the modal identification methods were compared with various model systems [5]. The latter method, ARMA model, is proposed to capture the critical modes of a power system from load switching which is the perturbations appear as noise with very small magnitudes on ambient data. This method uses the robust numerical techniques for calculating the oscillation modes. It uses the signal data in normal condition, hence it can perform faster. Additionally, the results from previous studies presented the accuracy and efficiency of this method [6]. Currently, the development of wide-area measurement system (WAMS) is growing rapidly [7]. System-wide dynamics of power systems can be synchronously measured and quickly centralized, which presents the potential of the online assessment of power dynamic characteristics [8]. In this paper, the accuracy and efficiency of methods for identifying online dominant oscillation modes in power system are investigated and compared. Multi-channel outputs such as the active power outputs of each generator are measured and transferred to assess oscillation modes by using ARMA model. The two solving algorithms in ARMA model which are used broadly are least-square and kalman filter algorithms [9]. These algorithms can be implemented
Chairerg Jakpattanajit, Naebboon Hoonchareon and Akihiko Yokoyama
in on-line measurement of low-frequency oscillations in power system. The structure of this paper is shown as follows: modal analysis and AMRA model are described in section 2. In section 3, two-area system case study and test procedures are presented. Summary of results is given in section 4.
the AR and MA terms, consecutively. The system transfer function Gs (q) between the output and the input is defined as:
Gs ( q ) =
B( q ) A(q)
(5)
p
A( q ) = 1 +
2. Modes of Oscillation Detectors
353
∑a q i
−i
(6)
i =0
d
B( q ) =
2.1 Modal Analysis
∑b q i
−i
(7)
i =0
The linearization technique around a steady state operating point is applied to identify oscillatory modes of a non-linear multi-machine power system. A dynamic system model is put into state space form [1][2]. Δx& = AΔx + BΔu
(1)
Δ y = C Δ x + DΔ u
The eigenvalue of the system, λI , is determined by characteristic equation of the state matrix A as below: det(λI − A) = 0 λ = λ1 , λ2 ,..., λn
(2)
For a particular eigenvalue, λi = σ i ± jωi , the real part of the eigenvalue gives the damping factor, and the imaginary part gives the frequency of oscillations. The corresponding damping ratio in percent is given by:
ξi =
−σ i
σ i2 + ωi2
× 100
(3)
A time series function can represent the power system linearized model as follows. y[ k ] = −
d
∑ a y[k − i ] + ∑ b e[k − i ] i
i =1
i
q is
the
shifting
operator
defined
by
q −1 y[k ] = y[k − 1] . 2.2.1 ARMA Model with the Least-Squares Algorithm The least-squares method is basically based on performing an optimization routine to obtain the unknown
[
parameters θ = a1 K a p
b1 K bd
]T
.
Unknown
parameters may be computed by minimizing an objective function such as
Min
V (θ ) =
1 N
N
1
∑ 2 ( y[k ] − yˆ[k ])
2
(8)
k =1
Where yˆ[k ] is the one-step-ahead prediction of y[k ] , and can be mathematically expressed as
yˆ[k | θ ] = ϕ T (k )θ
(9)
And the vector
2.2 System Identification with ARMA Model [3]
p
Where
(4)
ϕ (k ) = [− y(k − 1) K
− y(k − p) e(k − 1) K e(k − d )]
(10)
This requires an iterative search for θ that yields the minimum of loss function V (θ ) .
i=0
The output y[k ] and input noise e[k ] , which represent the load changing throughout the day, are related in a linear difference form. Where ai and bi represent the coefficients related to the autoregressive (AR) and moving average (MA) terms, respectively, and p and d denote the order of
2.2.2 ARMA Model with Kalman Filter Algorithm The kalman filter for estimating the state of unknown parameters θ from ARMA model is introduced in this section. Equation (11) is the discrete stochastic state-space model:
On-line Estimation of Power System Low Frequency Oscillatory Modes in Large Power Systems
354
x[k + 1] = F [k ]x[k ] + w[k ] y[k ] = H [k ]x[k ] + v[k ]
(11)
The unknown parameters, θ , can be calculated by transforming into (11). After the transformation, the unknown parameters are presented in (12).
Figure 1. This system is composed of four generators in two-area, 13 buses and two loads. The parameters of generators and transmission lines are referred from [1]. In this study, each generator is represented by fourth-order model and is equipped with governor model and excitation model.
G1
θ [k + 1] = θ [k ] + w[k ]
(12)
y[k ] = ϕ T [k ]θ [k ] + v[k ]
G11 3
1
101
13
11 110 12
10 T
F [ k ] = I , H [k ] = ϕ [k ]
When
2 20
The kalman filter can be applied for estimating the state of unknown parameters θˆ with following:
[
]
θˆ[k ] = θˆ[k − 1] + L[k ] y[k ] − ϕ T [k ]θˆ[k − 1] L[k ] =
(13)
P[k − 1]ϕ[k ]
(14)
R2 [k ] + ϕ T [k ]P[k − 1]ϕ[k ]
P[k ] = P[k − 1] −
P[k − 1]ϕ[k ]ϕ T [k ]P[k − 1] R2 [k ] + ϕ T [k ]P[k − 1]ϕ[k ]
[
T
]
+ R1[k ]
[
T
(15)
]
When R1[k ] = E w[k ]w [k ] and R2[k ] = E v[k ]v [k ]
The roots of polynomial A(q ) in (5) yield the poles of the system Z p = Z Rp + jZ Ip . They can be transformed onto the s-plane s p = σ p ± jω p using a bilinear transformation, i.e. s p =
2 (Z p −1) . When Ts is sampling interval in Ts (Z p + 1)
discrete time domain.
3. Case Studies This section consists of the two following parts: test system and test procedure. The first part describes the twoarea test system which is selected to be the representative of power system. The second part denotes the testing procedure of this study. 3.1 Test System The two-area test system used to study inter-area oscillation of an interconnected power system is shown in
G2
4 14
120 G12
Fig. 1. A two-area interconnected system. As described above, the test system has twenty-eight state variables. This test system comprises of the significant oscillation modes which are two local modes and one interarea modes. The oscillation modes depend on the operating point. 3.2 Test Procedures In order to investigate performance of least-square and kalman filter algorithms for on-line power oscillatory modes estimation, the accuracy and calculation time of both algorithms are compared. The process of on-line power oscillatory modes estimations is shown in Figure2. To calculate on-line power oscillation modes estimation, the first step is measuring and recording the system data which is the active powers of four generators and the sampling interval equals 10 msec. The next step is detrending the data to find out the differences between measured data and average data. The third step is filtering the data at frequency range 0 – 5 Hz. The forth step is resampling. The next step is calculating the unknown parameter ( θ ) from ARMA model by using least-squares or kalman filter algorithm. Then, the value of unknown parameter will be used for estimating the system poles. Finally, the results from the previous step will be used to calculate the dominant oscillation modes. To compare the performance of these two solving algorithms, the two-area model system is simulated for 60 minutes and the load is changed every 15 minutes. The oscillation modes both local and inter-area modes are shown in Table 1. The simulation was run on a PC with
Chairerg Jakpattanajit, Naebboon Hoonchareon and Akihiko Yokoyama
Core2 CPU 2.13 GHz and 3 GB RAM allocated to MATLAB program. Fig. 3 presents the sample of measured power of generator.
Gs (q) =
B(q) A(q)
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algorithm and the second subsection denotes the performance of the oscillation modes estimations using ARMA model with kalman filter algorithm.
Fig. 3. The sample of measured active power of generator with white noise. 4.1 ARMA model with least-square algorithm.
Fig. 2. On-line power oscillation modes estimations. Table 1. Test system condition and modal analysis
Load
Generation Inter-Area Mode Local 1 Mode Local 2 Mode
BUS4 (MW) BUS14 (MW) G1 (MW) G2 (MW) G11 (MW) G12 (MW) Damping (%) Freq. (Hz) Damping (%) Freq. (Hz) Damping (%) Freq. (Hz)
0-15 967 1767 700 700 719 700 0.972 0.622 7.772 1.126 8.009 1.092
Time (minute) 15-30 30-45 967 967 883.5 1943 834 682 350 770 360 791 350 770 7.0364 0.233 0.6383 0.604 11.313 5.937 1.0402 1.1396 20.517 7.235 1.002 1.096
45-60 967 1060 804 420 431 420 5.448 0.6413 10.940 1.052 17.491 1.037
In order to find out the performance of least-square algorithm, the window size (N) and the number of estimated poles (p) are changed in the calculation in term of autoregressive. Modes of oscillation are calculated every 1 second. The performance results are presented in term of maximum one step calculation times and root mean square error (RMSE) as shown in Table 2. Table 2. Performance of ARMA model with least-square algorithm Number Number of data of (N) estimated poles (p)
4
6
8
4. Test results This section describes the testing results of this study, which is divided into the two following subsections. The first subsection presents the performance of the oscillation modes estimations using ARMA model with least-square
400 600 800 400 600 800 400 600 800
one step cal. time (msec.)
329 446 574 383 491 623 457 555 682
RMSE Inter-Area Mode Local Mode Damping Freq. Damping Freq. (%) (Hz.) (%) (Hz.)
1.519 2.127 1.962 1.874 2.129 1.825 2.798 2.092 2.074
0.022 0.021 0.021 0.017 0.017 0.017 0.016 0.014 0.014
3.370 3.329 2.733 5.119 2.977 2.531 3.345 2.106 1.954
0.051 0.049 0.051 0.037 0.035 0.028 0.035 0.032 0.029
Figs. 4 and 5 depict the damping ratio and frequency of oscillations of inter-area mode. Figs. 6 and 7 demonstrate the damping ratio and frequency of oscillations of local mode. All four figures present the simulation results by using ARMA model with least-square algorithm.
On-line Estimation of Power System Low Frequency Oscillatory Modes in Large Power Systems
356
Inter-Area Mode Oscillation 20
15
15
10
10
Damping (%)
Damping (%)
Inter-Area Mode Oscillation 20
5 0 -5 -10 0
20 30 40 Time in minutes
50
-10 0
60
Fig. 4. percent damping of inter-area mode by using ARMA model with least-square algorithm (p=6 and N=800).
0 -5
modal analysis ARMA with least-square algorithm 10
5
modal analysis ARMA with kalman filter algorithm 10
Inter-Area Mode Oscillation
60
Inter-Area Mode Oscillation 0.75
0.7
0.7 Frequency (Hz.)
Frequency (Hz.)
50
Fig. 8. percent damping of inter-area mode by using ARMA model with kalman filter algorithm (p=6 and Δt =250).
0.75
0.65 0.6 0.55 0.5 0
20 30 40 Time in munites
20 30 40 Time in minutes
50
0.5 0
60
Fig. 5. oscillations frequency of inter-area mode by using ARMA model with least-square algorithm (p=6 and N=800).
0.6 0.55
modal analysis ARMA with least-square algorithm 10
0.65
modal analysis ARMA with kalman filter algorithm 10
20 30 40 Time in minutes
50
60
Fig. 9. oscillations frequency of inter-area mode by using ARMA model with kalman filter algorithm (p=6 and Δt=250).
Local Mode Oscillation 20
Local Mode Oscillation 20
15
Damping (%)
Damping (%)
15 10 5 0 -5 -10 0
modal analysis ARMA with least-square algorithm 10
20 30 40 Time in minutes
50
5 0 -5
60
Fig. 6. percent damping of local mode by using ARMA model with least-square algorithm (p=6 and N=800).
10
-10 0
modal analysis ARMA with kalman filter algorithm 10
20 30 40 Time in minutes
50
60
Fig. 10. percent damping of local mode by using ARMA model with kalman filter algorithm(p=6 and Δt =250).
Local Mode Oscillation Local Mode Oscillation
1.15
1.2
1.1
1.15
1.05
Frequency (Hz.)
Frequency (Hz.)
1.2
1 0.95 modal analysis ARMA with least-square algorithm
0.9 0.85 0
10
20 30 40 Time in minutes
50
1.1 1.05 1 0.95 modal analysis RAMA with kalman filter algorithm
0.9 60
Fig. 7. oscillations frequency of local mode by using ARMA model with least-square algorithm (p=6 and N=800).
0.85 0
10
20 30 40 Time in minutes
50
60
Fig. 11. oscillations frequency of local mode by using ARMA model with kalman filter algorithm (p=6 and Δt =250).
Chairerg Jakpattanajit, Naebboon Hoonchareon and Akihiko Yokoyama
4.2 ARMA model with kalman filter algorithm. The main difference between least-square and kalman filter algorithms is window size which the latter algorithm does not use for calculating the unknown parameter ( θ ). Thus, the performance of kalman filter algorithm can be tested by varying the number of estimated poles (p) in term of autoregressive and update time step (Δt). Table 3 presents the performances of these method in term of maximum one step calculation time and root mean square error (RMSE). These following figures present the simulation results by using ARMA model with kalman filter algorithm. Figures 8 and 9 depict the damping ratio and frequency of oscillations of inter-area mode. Figures 10 and 11 demonstrate the damping ratio and frequency of oscillations of local mode.
357
algorithm is less than the least-square algorithm because of less required number of data. Moreover, kalman filter algorithm can provide more accurate calculation results. Thus, ARMA model with kalman filter algorithm has more potential in online oscillation mode estimation and can be used to design a control system for enhancing the damping in a power system.
Acknowledgements This research was supported by the Faculty Development Scholarship of the Commission on Higher Education of Thailand with collaboration of AUN/SEEDNet.
References Table 3. Performance of ARMA model with kalman filter algorithm Number of estimated poles (p)
4
6
8
Δt (msec. )
one step cal. time (msec.)
100 250 500 100 250 500 100 250 500
12.5 18.5 23.2 18.2 19.9 23.7 23.9 24.4 25.4
RMSE Inter-Area Mode Damping Freq. (%) (Hz.)
Local Mode Damping Freq. (%) (Hz.)
3.624 2.012 2.091 4.465 2.095 2.462 5.108 2.365 37.126
2.268 3.172 14.056 2.925 1.890 42.220 3.147 2.796 14.953
0.047 0.015 0.014 0.026 0.015 0.018 0.023 0.013 0.146
0.042 0.100 1.072 0.048 0.036 0.863 0.034 0.042 0.589
5. Conclusion The oscillation modes in a power system can be estimated by using time domain data which is the active power of generator. In this study, ARMA Models with leastsquare and kalman filter algorithms are investigated and compared in the simulations of the two-area model system. The calculation results for estimating real time oscillation modes by these algorithms are satisfaction. Both algorithms can calculate the frequency of oscillation mode better than the damping ratio. The number of poles which is required in estimating the unknown parameter influences the calculation time of both algorithms. However, there are some significant distinctions. Firstly, the least-square algorithm requires a large amount of window size though they are not needed in kalman filter algorithm. Secondly, the calculation time of kalman filter
[1] [2] [3]
[4] [5]
[6]
[7]
[8]
[9]
P. Kundur, “Power System Stability and Control”, McGraw Hill, New York, USA, 1994. G. Rogers, “Power System Oscillations”, Kluwer Academic Publishers, Massachusetts, USA, 2000. J.F. Hauer, C.J. Demeure and L.L. Scharf, “Initial Results in Prony Analysis of Power System Response Signals”, IEEE Trans. on Power System, vol.5, no.1, pp.80-89, Feb. 1990. L. Ljung, “System Identification: Theory for the User”, Prentice Hall PTR, New Jersey, USA, 1999. J.J. Sanchez-Gasca and J.H. Chow “Performance Comparison of Three Identification Method for The Analysis of Electromechanical Oscillations”, IEEE Trans. on Power System, vol.14, no.3, pp.995-1002, Aug. 1999. R.W. Wies, J.W. Pierre and D.J. Trudnowski, “Use of ARMA Block Processing for Estimating Stationary Low-Frequency Electromechanical Modes of Power System”, IEEE Trans. on Power System, Vol.18, No.1, pp.167-173, Feb.2003 I. Kamwa, R. Grondin, and Y. Hebert “Wide-area measurements based stabilizing control of large power system”, IEEE Trans. on Power System, Vol. 16, pp. 136-153, 2001. K.M. EL-Naggar, “On-line Measurement of Lowfrequency Oscillations in Power Systems”, Measurement, vol.42, issue.5, pp.716-721, Jun. 2009. S.A. Soliman, A.M. Al-Kandari and M.E. El-Hawary, “Linear Kalman Filter Algorithm for Analysis of Transient Stability Swing in Large Connected Power Systems”, Electric Power System Research, vol.34, issue.3, pp.173-178, Sep. 1995.
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On-line Estimation of Power System Low Frequency Oscillatory Modes in Large Power Systems
Chairerg Jakpattanajit was born in Khonken, THAILAND, on September 28, 1975. He received B.Eng and M.Eng. degree in Electrical Engineering from Kasetsart University in 2000 and 2004. Currently, he is pursuing the Ph.D. degree at Chulalongkorn University. His research interests include FACTS, power system stability and control.
Naebboon Hoonchareon received the B.Eng. degree in Electrical Engineering from Chulalongkorn University, in 1993, the M.S. and Ph.D. degree in Electrical Engineering from Purdue University, in 1996 and 2000, respectively. Since 2000, he has been with the Department of Electrical Engineering, Chulalongkorn University. His research interest includes power system dynamics and control.
Akihiko Yokoyama was born in Osaka, Japan, on October 9, 1956. He received the B.S., M.S., and Dr. Eng. Degree from the University of Tokyo, Tokyo, Japan, in 1979, 1981, and 1984, respectively. He has been with Department of Electrical Engineering, the University of Tokyo, since 1984 and currently is a professor in charge of power system engineering. He is a member of IEEJ, IEEE, and CIGRE.
